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Writings by Uwe Lück

‘Representing Interval Orders by Arbitrary Real Intervals’

This solves a representation problem once left open by Fishburn and deals with some related themes.

Interval orders are a way to model non-transitive indifference in comparison judgments as well as temporal relations between events. While so far only characterizations of their representability by complete precedence vs. intersecting of closed or open (and bounded) real intervals have been known, this paper presents necessary and sufficient conditions for their representability by arbitrary real intervals as well as a new characterization for open representability. Furthermore we, like Fishburn, consider natural restrictions on representations and extend respective results of his. Accordingly, we also deal with semiorders in the sense of Luce. Interrelations of countability (separability, representability) conditions (“directly” in terms of interval orders) reveal two redundancies in Fishburn’s representability conditions and indicate more direct ways to his results. The key to our results is a pair of generalizations of Fishburn’s notion of “singularity”.

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Keywords: interval order, semiorder, linear order, representation; preferences/comparisons, economics/mathematical psychology/measurement.

Last update Mar 19, 2009

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